Time-domain analysis for predicting ship motions

T I M E - D O M A I N A N A L Y S I S FOR P R E D I C T I N G SHIP MOTIONS

Robert F. BECK a n d Allan R. M A G E E

Department of Naval Architecture and Marine Engineering, T h e University of Michigan, A n n Arbor, Michigan

The use of time-domain analysis f o r predicting ship motions is investigated. In the m e t h o d , the h y d r o d y n a m i c problem is solved directly in the time d o m a i n as a n initial value problem starting f r o m rest rather t h a n the m o r e conventional f r e q u e n c y - d o m a i n a p p r o a c h . For linearized problems t h e t i m e - d o m a i n a n d f r e q u e n c y - d o m a m results are Fourier t r a n s f o r m s of o n e a n o t h e r a n d are t h e r e f o r e c o m p l e m e n t a r y . For fully n o n l m e a r simulations the time-domain approach is prefen-ed.

In t h i s p a p e r both linear a n d b o d y - n o n l i n e a r problems will be d i s c u s s e d . T h e b o d y - n o n l i n e a r p r o b l e m r e q u i r e s t h e body b o u n d a r y condition to b e satisfied on t h e i n s t a n t a n e o u s position of t h e b o d y w h i l e maintaining the linearized free surface b o u n d a r y condition. In the linear p r o b l e m , both t h e free surface a n d body boundary conditions are linearized. T h e body boundary condition is linearized about the m e a n position of t h e body. Because the free surface condition is linearized about t h e calm w a t e r level, a time-domain G r e e n function approach is used to solve both p r o b l e m s .

Results of linear t i m e - d o m a i n c a l c u l a t i o n s are presented f o r t h e W i g l e y hull f o r m a n d c o m p a r e d w i t h experiments. Body-nonlinear computations are shown for a s u b m e r g e d ellipsoid. In both cases, t h e influence on t h e time-domain results of the singularity in t h e frequency domain at T = 1/4 is d i s c u s s e d .

1. I N T R O D U C T I O N

T i m e - d o m a i n a n a l y s i s h a s p r o v e n t o b e a useful and e n l i g h t e n i n g t o o l t o a n a l y z e s h i p m o t i o n s . In the m e t h o d , t h e h y d r o d y n a m i c p r o b l e m is s o l v e d directly in t h e t i m e d o m a i n a s a n i n i t i a l - v a l u e Drcblem. Its simplicity a n d applicability t o a w i d e ^/ariety of p r o b l e m s a r e m a j o r a d v a n t a g e s . T h e Domputational a l g o r i t h m s r e m a i n s u b s t a n t i a l l y t h e same r e g a r d l e s s of t h e a m p l i t u d e , d i r e c t i o n or t i m e l i s t o r y of t h e m o t i o n . T h e m a j o r d i s a d v a n t a g e is :hat t h e solution must be t i m e - s t e p p e d w i t h m e m o r y n t h e s y s t e m . T h i s c a n l e a d t o n u m e r i c a l n s t a b i l i t i e s f o r a f u l l y n o n l i n e a r a p p r o a c h , a n d ' e q u i r e s t h e e v a l u a t i o n of c o n v o l u t i o n i n t e g r a l s in :he G r e e n f u n c t i o n a p p r o a c h . T h e c o n v e n t i o n a l a p p r o a c h t o s o l v i n g s e a k e e p i n g p r o b l e m s is t o d e v e l o p a f r e q u e n c y -d o m a i n s o l u t i o n . F o r l i n e a r i z e -d p r o b l e m s a t : o n s t a n t o r zero fonward s p e e d t h e t i m e - d o m a i n a n d r e q u e n c y - d o m a i n s o l u t i o n s a r e F o u r i e r t r a n s f o r m s Df o n e a n o t h e r a n d a r e , t h e r e f o r e , c o m p l e m e n t a r y . Dne m e t h o d o r t h e other m i g h t h a v e a d v a n t a g e s for a particular p r o b l e m .

At t h a present t i m e , fully n o n l i n e a r t i m e - d o m a i n solutions f o r arbitrary t h r e e - d i m e n s i o n a l b o d i e s are j n d e r d e v e l o p m e n t b u t a r e n o t y e t p r a c t i c a l , r h e r e f o r e , a n i n t e r m e d i a t e a p p r o x i m a t i o n , t h e s o -j a l l e d N e u m a n n - K e l v i n a p p r o a c h , h a s b e e n u s e d )y m a n y r e s e a r c h e r s . In t h e N e u m a n n - K e l v i n a p p r o a c h t h e f l u i d is c o n s i d e r e d i n c o m p r e s s i b l e a n d inviscid so that t h e L a p l a c e e q u a t i o n g o v e r n s the flow. T h e body b o u n d a r y condition is applied on the exact b o d y s u r f a c e , but a linearized free surface b o u n d a r y c o n d i t i o n is u s e d . T h e s e a s s u m p t i o n s a l l o w t h e d e v e l o p m e n t of n u m e r i c a l s o l u t i o n t e c h n i q u e s using a G r e e n function a p p r o a c h . T h e s o c a l l e d p a n e l m e t h o d s h a v e b e e n u s e d o n a v a r i e t y of p r o b l e m s . A t z e r o f o r w a r d s p e e d t h e f r e q u e n c y - d o m a i n p a n e l m e t h o d enjoys w i d e s p r e a d popularity in t h e o f f s h o r e industry. T h e c o n f i d e n c e level in t h e results is quite h i g h , (see f o r e x a m p l e K o r s m e y e r et al., 1988). At steady f o r w a r d s p e e d , t h e f r e q u e n c y - d o m a i n p a n e l m e t h o d e n c o u n t e r s d i f f i c u l t i e s b e c a u s e t h e G r e e n f u n c t i o n is c o m p l i c a t e d a n d difficult t o c o m p u t e . N e v e r t h e l e s s ,

results h a v e b e e n o b t a i n e d by several researchers i n c l u d i n g C h a n g ( 1 9 7 7 ) . Inglis a n d Price ( 1 9 8 1 ) , a n d G u e v e l a n d B o u g i s (1982).

T h e u s e of t i m e - d o m a i n methods is not new. T h e s o l u t i o n f o r t h e f u n d a m e n t a l 1 / r s i n g u l a r i t y is credited to Finklestein ( 1 9 5 7 ) . Discussions of direct t i m e - d o m a i n s o l u t i o n s a r e p r e s e n t e d by v a r i o u s a u t h o r s s u c h a s : S t o k e r ( 1 9 5 7 ) , C u m m i n s ( 1 9 6 2 ) . O g i l v i e ( 1 9 6 4 ) , a n d W e h a u s e n ( 1 9 6 7 ) . A s c o m p u t a t i o n a l p o w e r h a s i n c r e a s e d , it h a s b e c o m e p r a c t i c a l t o s t u d y a c t u a l s o l u t i o n s a n d i n v e s t i g a t e t h e c o m p u t a t i o n a l a d v a n t a g e s of t i m e - d o m a i n m e t h o d s . A d a c h i a n d O m a t s u ( 1 9 7 9 ) , Y e u n g ( 1 9 8 2 ) , N e w m a n ( 1 9 8 5 ) , B e c k a n d Liapis ( 1 9 8 7 ) ,

K o r s m e y e r (1988), Korsmeyer etal. (1988), King et al. (1988), a n d Ferrant (1989) are among those who have successfully obtained results.

For linear problems at zero forward s p e e d , the t i m e - d o m a i n computations are not a s fast as t h e c o n v e n t i o n a l frequency-domain approach because m a n y t i m e steps are n e e d e d (rather t h a n a f e w frequencies) to obtain a n adequate representation of t h e results. H o w e v e r , at f o r w a r d s p e e d t h e f r e q u e n c y - d o m a i n G r e e n function b e c o m e s very difficult t o c o m p u t e a n d the t i m e - d o m a i n method a p p e a r s to be significantly faster. For problems w h e r e the body boundary condition is applied on t h e i n s t a n t a n e o u s exact body surface the time-d o m a i n m e t h o time-d is the only alternative; frequency-d o m a i n solutions are limitefrequency-d to a few simple cases.

In this paper the theoretical development for the t i m e - d o m a i n approach given arbitrary body shape a n d motion will be presented. T h e reduction of the formulation to the case of linear motions at constant f o r w a r d s p e e d will also be outlined. Comparisons with strip theory a n d experiments will be given. The

influence of t h e singularity a r o u n d T = =-7 , 8 4 w h e r e UQ = the body velocity, cog is the encounter f r e q u e n c y a n d g is the acceleration of gravity, will also be discussed.

2. THEORETICAL FORMULATION 2 . 1 . Radiation Forces

The origin of the axis system is fixed on the free surface of an infinitely d e e p , incompressible, ideal fluid, initially at r e s t T h e z-axis is positive upwards a n d the x - y plane corresponds to the calm water l e v e l . T h e direction of m o t i o n of t h e body is generally in the positive x direction. T h e governing e q u a t i o n in t h e fluid f o r t h e velocity perturbation potential is the Laplace equation:

w h e r e V'^<t>ix,y,z,t) = 0 U =

V4>

T h e body b o u n d a r y c o n d i t i o n is satisfied o n the instantaneous position of the body surface:

dn

n o n S/,(t)- (4)

w h e r e n = i n w a r d unit n o r m a i t o t h e b o d y surface, out of the fluid.

V = instantaneous velocity of a point o n the body surface including a n g u l a r velocity effects.

At infinity the fluid velocities must all go to zero s u c h that:

0 as - > ~ (5) a n d

V ^ - » 0 as r - 4 - 0 0 (6) The initial conditions are that:

0 ^ 0

dt J

as t-*-oo (7)

In the usual manner, a n integral equation t h a t must be solved to determine 0 is found by applying Green's theorem to t h e fluid domain yielding:

'dvUv2G-GV2<&)= [ds[<t>~G^\ (8

J \ 1

i

V S

w i t h t h e v o l u m e V b o u n d e d b y S , w h e r e S = Sf,+Sf + S„ a n d Sh= body s u r f a c e , 5 / = free surface, and S«,= surface at infinity.

T h e G r e e n f u n c t i o n f o r t h e t i m e d e p e n d e n t problem is given by \r r J OQ G(P.(2,f,T) = J ^ V ^ s i l l ( V ^ ( f - T ) ) € * ^ ^ - ^ ^ ^ ( « ? ) P = {x{fiy(t\i{t)) Q = (I(T).TI(T).C(T)) r = [ { x - ^ f + ( y - 7 , f - H ( z - C ) ' r ' = [ ( x - 4 f + ( y- 7 7) ^ + ( z + C) r -jll/Z R = \ x - ^ f ^ { y - ^ f \

5 ( 0 = delta function where \5{f)f{t)dt = f{Qi_

1/2

il/2 (9]

H{t)= unit Step function = 0 r < 0

It is easily shown that t h e G r e e n function has t h e following properties: V^G = ^7iS(P-Q)5{t-x) d^G^ do ^ _ — = r + g - r - = 0 o n z = 0 dr dz G,^ = 0 t-x<0 dt V C -> 0 r - > «

As s h o w n in Magee (1990), substituting the Green function into (8), integrating both sides with respect to X a n d then reducing t h e integrals over the free s u r f a c e u s i n g S t o k e s ' T h e o r e m results in t h e following integral equation for 4>(P,t):

Sh(') ~ h J J ' ^ c { ^ ( Ö . ' f ) ^ G{P,Q,t,x) — Shit) (10) -G{P,Q,t,x) drtQ ^(ö.-r) ~ 1 j'^Q { ^ ( Ö . T ) ^ G{P,Q,t,x) - G ( P . Ö , r , T ) ^ 0 ( Ö . T ) | v ^ ( C , T )

where r ( t ) is the curve d e f i n e d by the instantaneous intersection of t h e m e a n hull position a n d t h e z=0 plane a n d V/^ i s t h e t w o - d i m e n s i o n a l n o r m a l velocity in the z=0 plane of a point on r. It should be n o t e d that in t h e l i n e a r i z e d p r o b l e m t h e line integrals are zero at zero f o r w a r d speed and reduce t o t h o s e given b y King et al. (1988) f o r constant f o r w a r d s p e e d . F o r t h e m a n e u v e r i n g c a s e of unsteady, large amplitude motion in t h e z=0 plane, (10) reduces to t h e equation given by Liapis (1986), A p p e n d i x A . F o r t h e b o d y - n o n l i n e a r p r o b l e m t h e line integrals have n o n z e r o v a l u e s e v e n at zero f o r w a r d s p e e d unless t h e body is wall sided for all points on r for all time.

In m a n y situations a s o u r c e formulation is more convenient because it leads t o easier computations for the tangential velocities o n t h e body surface. In the usual manner of potential theory, it is possible to

derive t h e following integral equation for t h e s o u r c e strength on t h e body s u r f a c e : - o ( P . O 1 2 ^"c/\ dnp\r r ) dnp + T - f ^ ^ 11 ^Q''')T-GiP,Q,t,x) ATC J

.

where V„ is the three-dimensional normal velocity of a point on T(t), a(P,t) is t h e u n k n o w n source s t r e n g t h , a n d the potential on t h e body surface is given by <l>iP,t) = ~ f f dSQoiQj -Shi') - T - f'^^ f f '^Q o{Q,x)G{P,Q,t,x) An J ii ^ — Shi^) ₍₁₂₎ \dx ^dtQ a(Ö.T)V„(Ö,T) 47Cg J i ^ r ( r ) Vf^{Q,x)GiP,Q,t,x)

T h e hydrodynamic forces acting o n t h e body due to a prescribed body motion are found by integrating the pressure over the body surface. Neglecting the hydrostatic pressure, t h e u n s t e a d y hydrodynamic pressure is given by Bernoulli's e q u a t i o n :

- = - f — + V - v V - - V ^ p - i - V - V ^ (13)

p Kdt ) 2

T h e g e n e r a l i z e d f o r c e o n t h e b o d y in the y t h direction is then given by:

dSPni (14) Shi') Where nj , r e p r e s e n t i n g t h e g e n e r a l i z e d unit normal, is defined a s («l.'»2'''3) = '' (n4,«5,rt6) = r x n r = ( J , 7 , z )

{x,y,z)= body axis system

a n d y = 1,2,3 corresponds to the directions of t h e xj,z axes, respectively.

The above formulation is for the body-nonlinear p r o b l e m in. w h i c h the b o d y boundary condition is satisfied on t h e instantaneous position of the body surface. Because of this, the linear system theory normally invoked in seakeeping analysis cannot be u s e d . T h e b o d y - n o n l i n e a r a p p r o a c h is primarily useful for nonlinear simulations.

A linear time-domain analysis may be developed for the c o n s t a n t fon^vard s p e e d case a n d can be c o m p a r e d directly with f r e q u e n c y - d o m a i n analysis. Either impulsive or nonimpulsive input can be u s e d . Liapis a n d Beck (1985) d e v e l o p e d a theory for the i m p u l s i v e radiation p r o b l e m . King etal. (1988) d e v e l o p e d a n o n i m p u l s i v e a p p r o a c h to both t h e r a d i a t i o n a n d e x c i t i n g - f o r c e p r o b l e m s . By a p p r o p r i a t e c h o i c e of t h e n o n i m p u l s i v e i n p u t , numerical errors in the computation can be reduced.

In l i n e a r t i m e - d o m a i n a n a l y s i s it is m o r e convenient to w o r k in a coordinate s y s t e m fixed t o the moving v e s s e l . In this c a s e , the total velocity potential is defined a s :

(15) OJ- {x,y, z,t) = -UQX + OQ{x,y, z) + ^Q{x,y, z,t) + ^{x,y,z,t)

where

-UQX + OO = potential due to steady translation ^0 = incident wave potential

normal given in (14) and , resulting from the steady forward motion, is given by

{mi,m2,mi) = -{n-V)W

(m4,m5,*^) = - ( n - V ) ( r x W )

(.|-W = V{-UQX + ^Q)

W is the fiuid velocity due to t h e s t e a d y fopA/ai motion of the v e s s e l in the ship f i x e d c o o r d i n a f r a m e . T h e linearized free s u r f a c e c o n d i t i o n written a s :

^-UQ4-\ h + 8^<l>k=^ on z = 0 (1< \ot ax J az

The initial conditions for the unsteady potentials an

d<t>k dt 5.0

S as t ^ ' k = l,2 7 _(1i

Since the d i s t u r b a n c e s g e n e r a t i n g t h e unsteac p o t e n t i a l s originate in t h e n e i g h b o r h o o d of It origin * ( x , y , z , t ) = S h{^,y,z,t) /fc = 7 is the diffracted w a v e k = l,2,.-,6 are t h e p o t e n t i a l s due t o t h e b o d y m o t i o n s s u r g e , s w a y , h e a v e , roll, pitch a n d y a w , r e s p e c t i v e l y . T o m e e t t h e a p p r o p r i a t e b o d y b o u n d a r y c o n d i t i o n , ^ ^ = 0 o n t h e m e a n b o d y dn

surface SQ , t h e following b o u n d a r y conditions a r e specified for the various potentials:

V<l>k{x,y.z.t)^0 on Seo /k = 1.2,...,7 (2( dn

ih.

It can be s h o w n that the Green function given by (! also satisfies these conditions if t h e approprial t r a n s f o r m a t i o n is made to t h e m o v i n g coordinai system.

In developing the linear boundary value proble implied by equations (15) t h r o u g h (20), there is £ implicit assumption that the body g e o m e t r y is sue that the s t e a d y disturbance potential 0Q is s m a This is a consequence of the free surface bounda condition that has been linearized a r o u n d the fre s t r e a m velocity UQ- In addition, t h e amplitudes i motion must be small b e c a u s e t h e body b o u n d a c o n d i t i o n has b e e n e x p a n d e d a b o u t t h e mes position of the body surface.

A s with the body-nonlinear p r o b l e m , an integr equation t o determine the u n k n o w n linear potentia is found by applying Green's t h e o r e m a n d using tl" Green function (9). The final result m a y be found King etal. (1988) and is

w h e r e Cjfc(t) is the displacement in the 1^^ mode of m o t i o n , and the overdot represents the derivative with respect to time, njc is the generalized unit

(21) So ^ t r driQ t f . —00 r - G ( P . e . r - T ) ^ 0 ( ö , T ) -uJ^^[Q,x)j- G{P,Q,t-x) -G{P,Q,t-x)j-ip{Q,x)

where r is the curve defined by the intersection of the m e a n hull position a n d the z = 0 p l a n e .

In t h e l i n e a r c a s e (13) a n d (14) m a y be linearized to y i e l d :

Fjki.') = - P j j + W'- J n , . (22) So

T h e gradient of may be eliminated from (20) by e m p l o y i n g a t h e o r e m d e r i v e d by T u c k (cf. Oqilvie, 1 9 7 7 ) :

^ J dS[mj(pk + nj{W•'V(t>k)] =-jcU (p^njilxn)'.W

SQ r (23)

w h e r e / is the tangent v e c t o r to the wateriine curve r . Substituting (23) into (22) results in

SQ SQ

+pjcU(p,,nj(lxn)-W

(24)

It should be noted that since W is tangential to the ship hull it is almost perpendicular to {Ixn) and the line integral term is of higher order. If W is parallel to the wateriine this is exactly true a n d the line integral is zero. For the results shown in this paper t h e c o n t r i b u t i o n of t h e line i n t e g r a l to t h e h y d r o d y n a m i c forces acting on the body will be neglected.

For the linear problem, the hydrodynamic forces acting on the vessel c a n be related to the more t r a d i t i o n a l f r e q u e n c y - d o m a i n a d d e d m a s s a n d d a m p i n g . T h e radiation force in mode j due to a n imposed motion in mode k may be written in general as (cf. Cummins, 1962):

Pjki') = -fijk^ki') - bjktki^) - CjkCk(t)

'dxKj„it-x)Ux)

(25)

where Kjj^ represents the memory effect due to the free surface a n d the hydrostatic restoring forces have been neglected.

It is shown in King (1987) that bjk is zero, cy^ is a h y d r o d y n a m i c force that is proportional to the body displacement and is given by

cyJt = - P J j < i 5 hcoMj

-pjdl

W

where'^j^oo represents the large-time limit of the potential (pf.. (pj^ has a non-zero large-time limit

because - ^ f ^ = mjt w h i c h is nonzero. T h e ' ~* an

integral equation that must be solved to determine ^jtcw is given in King ef al. (1988).

T h e factor represents the infinite frequency a d d e d mass a n d is given by

l^jk=p\\^Wkf'j

So

(27)

where y/^ >s the solution to the integral equation

SQ SQ

For sinusoidal m o t i o n s , the radiation forces are usually g i v e n in tei-ms of t h e a d d e d m a s s a n d damping coefficients. T h e equivalence between the t i m e - d o m a i n force f o r m u l a t i o n a n d the f r e q u e n c y

domain is f o u n d by substituting ?jfe(f) = e " " into (25) a n d e q u a t i n g it w i t i i t t i e f r e q u e n c y - d o m a i n representation:

icot

= [ca^Ayjt (<u) - icoBjkico)y'"

Equating real and imaginary parts yields o o

Ajki^) = lijk ƒ ^ / t ( O s i n t u r 0

o o

Bji^ico) = dco Kjk{t)co%cot

(28)

(29)

T h e t e r m cji^ is f r e q u e n c y i n d e p e n d e n t . Since it multiplies the motion amplitude, it could be added to the hydrostatic restoring force c o e f f i c i e n t s in . t h e usual frequency-domain equations of motion. In the typical strip theory (cf. Salvesen, T u c k , a n d Faltisen, 1970 ) this term does not appear. W h e n comparing t i m e - d o m a i n p r e d i c t i o n s w i t h s t r i p t h e o r y it, therefore, appears reasonable to retain cyj^with the a d d e d m a s s . In e x p e r i m e n t s , the h y d r o d y n a m i c part of t h e s p r i n g c o n s t a n t t e r m is s o m e t i m e s s u b t r a c t e d off and at other t i m e s it is not. T h u s , w h e n c o m p a r i n g t i m e - d o m a i n p r e d i c t i o n s w i t h experiments one must be careful. T h e influence of this t e r m on the t i m e - d o m a i n p r e d i c t i o n s will be shown in section 4.

2.2. Exciting Forces

Consistent with the radiation force p r o b l e m , the exciting f o r c e s f o r the b o d y n o n l i n e a r N e u m a n n -Kelyin problem are c o m p u t e d assuming the incident w a v e system is linear. T h e boundary condition o n the free surface is still g i v e n by (3) but the normal velocity on t h e body is m o d i f i e d t o i n c l u d e t h e induced velocity of the incident w a v e s y s t e m . A s s h o w n in King etal. (1988) the i n d u c e d velocity o n the body is given by

(30) It i cos /3 sin /? k i where c j = x c o s ^ + ysin;3

Co(0 = arbitrary incident wave amplitude at origin P = wave propagation angle {n = head seas)

The function K m a y be identified as the impulse response function for the velocity field in the fluid resulting from an impulsive wave elevation at the origin. The w a v e s in (30) are long c r e s t e d ; short crested seas could be simulated by adding w a v e s propagating in different directions. T h e required body b o u n d a r y c o n d i t i o n for the b o d y - n o n l i n e a r problem is then given by

dn (31)

The remaining boundary conditions on the velocity potential (5) - (7) are unaltered. T h e r e f o r e , t h e c o m p u t e r c o d e that s o l v e s the radiation p r o b l e m can be easily modified to include the exciting force problem. A simple addition of the s e c o n d t e r m in (31) to the body boundary condition is all that is necessary.

King etal. (1988) give the solution of the exciting force problem in the linear case at both zero a n d non-zero forward s p e e d .

3. N U f ^ E R I C A L lülETHODS

The principal numerical task in the t i m e - d o m a i n m e t h o d is to s o l v e the integral e q u a t i o n for t h e perturbation potential ((10) or (21)) or the s o u r c e strength (11). T h e hydrodynamic forces acting on the body are t h e n f o u n d from the integration ol pressure over the body surface at each time step so t h a t in t h e f u t u r e , w h e n full s i m u l a t i o n s are attempted, these forces will be available to solve the equations of motion. To solve the integral equation a panel method is u s e d in which the instantaneous w e t t e d s u r f a c e of t h e b o d y is d i v i d e d inte quadrilateral panels. O n each panel t h e u n k n o w r potential or source strength is assumed constant. / t i m e m a r c h i n g s c h e m e is u s e d t o s o l v e the governing integral equation in the time d o m a i n . A e a c h t i m e s t e p a n e w v a l u e of t h e u n k n o w n is d e t e r m i n e d on e a c h p a n e l . T h e c o n v o l u t i o r integrals over the time history are evaluated using e simple trapezoidal rule. T h e details of the numerica method may be f o u n d in fvlagee (1990) for the b o d y n o n l i n e a r p r o b l e m o r K i n g (1987) for t h e l i n e a p r o b l e m .

For the linear problem, the calculations of t h e m t e r m s (17) are d o n e using two methods. T h e firs (and most c o m m o n ) is to use simplified m f s in w h i c i they are c o m p u t e d assuming W = - U o I . In this case the simplified mj's are given by:

m;=(0,0,0,0,C/o"3.-'^0''2) (32) The s e c o n d technique is to calculate the mj's using a finite difference s c h e m e . T h e required t e r m s are d e r i v e d f r o m t h e n o r m a l derivatives of the fluid velocities due to the steady flow on the body surface (cf. Ogilvie, 1977). T h e value of the fluid velocities on the body a n d in the fluid a small distance. A n , away f r o m the body along the normal vector c a n easily be calculated from the known solution to the steady N e u m a n n - K e l v i n problem (cf. Doctors a n d Beck, 1987). O n c e t h e s e velocities are k n o w n , a simple finite difference can be used to determine the r e q u i r e d n o r m a l d e r i v a t i v e s , a n d these c a n be extrapolated to the limit as An->0.

T h e p r o c e d u r e w o r k s quite well for the s e m i -infinite f l u i d p r o b l e m of t h a d o u b l e - b o d y f l o w . However, when the w a v e terms are included in the computations for the gradients of the velocities, the results are highly oscillatory, and do not c o n v e r g e for this finite difference s c h e m e . A s expected, it is the p a n e l s near the free surface that c a u s e the difficulties. T h e r e f o r e , in s e c t i o n 4 results are presented only for simplified and double body mj's.

T h e m a j o r p a r t of t h e c o m p u t e r t i m e is c o n s u m e d e v a l u a t i n g t h e G r e e n f u n c t i o n . In a typical m n of the body-nonlinear program order NP2 X KT2 G r e e n f u n c t i o n e v a l u a t i o n s are r e q u i r e d , w h e r e N P is the n u m b e r of panels and K T is the n u m b e r of t i m e s t e p s . To e v a l u a t e the G r e e n function efficiently, a vectorized technique has been d e v e l o p e d by f ^ a g e e a n d Beck (1989). Integrals involving the 1/r terms are performed using methods similar to Hess a n d Smith (1964). To evaluate the wave terms they are first written in a two-parameter, nondimensional f o r m : o o where (33) P = ix,y.z) X = kr' r ' = [ ( x - | ) 2 + ( y - r , ) 2 + ( z + c f

T h e parameter n relates the vertical to horizontal distance between source a n d field points, and p is time-like a n d relates to the phase of the generated waves. The function is oscillatory for large p and is sharply peaked (though not singular) for fi near 0. A plot of G is given in figure 1 .

A b i c u b i c interpolation t e c h n i q u e has b e e n d e v e l o p e d to compute G quickly using a vector processor. To reduce m e m o r y requirements a n d increase the accuracy of the interpolation, a t w o -step approach is taken. Rrst, (33) is written as

w h e r e

C(/i,)3) = exp G{0,p) + Gi{fi,p)

(34)

is given in Wehausen a n d Laitone (1960) and is the value of the Green function when both the source a n d field points lie on the free surface (i.e., ^ = 0 ) . T h e f u n c t i o n G{0,p) m a y be p r e c o m p u t e d a n d stored for simple one-dimensional interpolation. T h e s e c o n d s t e p is an interpolation of t h e G,(/i,)3) function in / i a n d p space. Figure 1 also s h o w s a plot of Gi{^,p). Note that G,(^,/J) is s m o o t h a n d a small percentage of G{n,p), thus allowing a much c o a r s e r g r i d s p a c i n g for e q u i v a l e n t a c c u r a c y . Gi{^,p) is interpolated using bicubic interpolation o n a nonuniform grid spacing. On a C R A Y Y-I^P the G r e e n function routine runs at approximately 140 M i l l i o n F l o a t i n g Point O p e r a t i o n s p e r S e c o n d ( M F L O P S ) a n d t a k e s a p p r o x i m a t e l y 2 . 7 m i c r o s e c o n d s per call f o r G a n d its derivatives including t h e return to physical space for a vector length of 2 0 0 .

In order to understand the difficulties associated w i t h t i m e - d o m a i n a n a l y s i s , it is i n s t r u c t i v e to examine the form of the potential d u e to an isolated source traveling below the free surface. Figures 2a a n d 2b s h o w the time history of the potential and its t i m e derivative for a field point travelling o n the free surface located a horizontal distance R = 1 behind the source point. The source is travelling at a depth of z = -2 b e l o w the free surface w i t h a Froude number, F„ = U Q / ^ , equal to 0.4. In this case, the

critical w a v e number w h e r e T = 1/4 is kR = .3906. T h e source has unit strength and is created at t = 0; t h e r e a f t e r , its s t r e n g t h r e m a i n s c o n s t a n t . T h e potential has the form of a typical impulse response

1.5¬ ©

-|\

Figure 2. The potential and Its time derivative

for a unit source. Undamped (a=0.0); Damped (a=0.05).

Figure 3. Greatly expanded view of the large-time tail of (a) the potential and; (b) its time derivative for the source of figure 2.

a=0.0; 0=0.05.

0.0

- 1 , 6

K R

Figure 4. Fourier transform of the time derivative of the potentials of figure 2b.

0=0.0;

0=0.05; Frequency domain Green function by Wu and Eatock Taylor, 19

function: tiiere is an initial large response followed a decaying oscillation about the large-time limit.

T h e s o l i d lines in f i g u r e s 3 a a n d 3 b a r e e x p a n d e d views of the oscillatory tail at large t i m e . T h e oscillations are at a frequency e q u i v a l e n t to T = 1/4 a n d lead to the singular b e h a v i o r in the frequency domain. Recall that the Fourier transform of a sine wave of frequency COQ in the time domain is a delta function at cog in the frequency d o m a i n . T h u s , the x = 1/4 singularity appears as a decaying sine w a v e at the appropriate frequency in the large-time tail of the large-time-domain results.

Figure 4 shows the real a n d imaginary parts of the F o u r i e r t r a n s f o r m of t h e d e r i v a t i v e of t h e potential (figure 2b). T h e Fourier transform of the time derivative is equivalent to the v a l u e s in the f r e q u e n c y d o m a i n of t h e G r e e n f u n c t i o n f o r a translating, pulsating source. T h e small s q u a r e s in figure 4 are the values of the G r e e n function taken from W u and Eatock Taylor (1987). As can be s e e n , t h e a g r e e m e n t b e t w e e n t h e f r e q u e n c y - d o m a i n computations and the Fourier transform of the t i m e -d o m a i n r e s p o n s e is e x c e l l e n t . T h e s i n g u l a r behavior of the frequency-domain Green function at X = 1/4 is clearly visible.

T o eliminate the a n o m a l o u s behavior in t i m e -d o m a i n simulations -d u e to the T = 1/4 singularity, artificial d a m p i n g c a n be introduced into the large-t i m e large-t a i l of large-the large-t i m e - d o m a i n G r e e n f u n c large-t i o n . In particular, the large t i m e a s y m p o t i c e x p a n s i o n (cf. K i n g , 1 9 8 7 ) that is u s e d in the G r e e n f u n c t i o n subroutines is multiplied by e " " ^ ^ " ^ ^ ) ^ w h e r e a is an arbitrary constant which determines the strength of the d a m p i n g . Typical values of a are in the range 0.0 to 0.1 T h e artificial d a m p i n g has the effect of forcing the large-time tail of the time-domain G r e e n function t o decay exponentially to zero. T h e d a s h e d c u r v e s in f i g u r e s 2 , 3 , a n d 4 are t h e results c o m p u t e d u s i n g t h e d a m p i n g . In f i g u r e 2 t h e d a s h e d curve cannot be s e e n because on the scale of the figure the d a m p e d and u n d a m p e d curves are identical. T h e effects of t h e d a m p i n g are clearly seen, however, in the expanded scale of figure 3. In this figure, the oscillations in the d a m p e d curve are substantially lower than in the u n d a m p e d c a s e a n d they d e c a y much faster.

T h e e f f e c t s of t h e d a m p i n g in the f r e q u e n c y d o m a i n c a n be seen in figure 4 . T h e d a m p e d a n d u n d a m p e d c u r v e s are effectively identica except around x = 1/4. In this region, the damping s m o o t h s out the singularity. A s t h e d a m p i n g coefficient a is r e d u c e d , t h e r e s u l t s b e c o m e m o r e a n d m o r e singular. In section 4 , the effects o f t h e d a m p i n g on motion simulations will be s h o w n .

4. R E S U L T S

A n e x a m p l e of c o m p u t e d results using linear t i m e - d o m a i n analysis is s h o w n in figures 5 a n d 6. T h e s e f i g u r e s s h o w the h e a v e a n d pitch a d d e d

mass and damping as a function of frequency for a modified Wigley mathematical hull form. The Wigley form has a lengthtobeam ratio of 10 a n d a b e a m -to-draft ratio 1.6. T h e half b e a m is g i v e n by t h e e q u a t i o n :

^ = 1 - U l - ( ^ 1 - H 0 . 2 ( — I

\ T ) y \ T ) j\ \ L )^

w h e r e b = the half beam of the model L = model length

T = model draft

For the calculations, 240 panels (30 lengthwise x 8 girthwise) were u s e d on the halfbody. The n o n -dimensional time s t e p , d e f i n e d as At'=At.^j-, w a s 0.088 a n d the total number of time s t e p s was 2 5 6 . Figure 5 presents results for zero forward speed a n d figure 6 is for a Froude number of 0.3. Also s h o w n in the figure are strip theory results c o m p u t e d using a S a l v e s e n , T u c k , Faltinsen (1970) b a s e d s t r i p theory (cf. Beck, 1989). T h e experimental results are d u e to G e r r i t s m a ( 1 9 8 8 ) . No e x p e r i m e n t a l results are available for Fn=0.0.

T h e W i g l e y hull is f o r e - a n d - a f t s y m m e t r i c . C o n s e q u e n t l y , f o r Fn=0.0 (figure 5) t h e c r o s s coupling coefficients are z e r o . A s can be seen in the figure 5, the strip theory a n d time-domain results a g r e e very well for high f r e q u e n c i e s . For low f r e q u e n c i e s , h o w e v e r , t h e r e a r e s i g n i f i c a n t d i f f e r e n c e s . T h i s is p r e s u m a b l y d u e to t h r e e -d i m e n s i o n a l e f f e c t s t h a t are n e g l e c t e -d in s t r i p t h e o r y . Note t h e o s c i l l a t i o n s in t h e d a m p i n g coefficients in the frequency range 6. - 8. which are a result of irregular f r e q u e n c i e s . Inglis and Price (1981) used a box barge with the same dimensions as the ship to estimate the irregular frequencies. By this m e t h o d , the first irregular frequency occurs at 5.7 and the s e c o n d at 7.9. T h e singular nature of the coefficients a r o u n d the irregular frequencies is not properly c a p t u r e d b e c a u s e t h e record length and time step size are insufficient to give adequate resolution to the Fourier t r a n s f o r m . T h i s is both a w e a k n e s s and strength of the t i m e - d o m a i n m e t h o d . As with the x = 1/4 singularity, in the time domain the irregular frequencies a p p e a r as oscillations in t h e large-time tail a n d d o not c a u s e a n y p a r t i c u l a r p r o b l e m s until t h e F o u r i e r t r a n s f o r m is t a k e n . Because of the finite record length a n d t i m e - s t e p size, the irregular frequencies are effectively filtered o u t O n the other h a n d , in the frequency domain t h e results are singular a n d calculations in the region of the irregular frequency are meaningless.

Figure 6 presents the results at forward s p e e d , it should be noted that the added m a s s coefficients include the Cj^/toS term as presented in (29). This is consistent with strip theory and Gerritsma's (1988) e x p e r i m e n t a l r e s u l t s s i n c e he u s e d o n l y t h e hydrostatic coefficients in reducing t h e m e a s u r e d d a t a t o o b t a i n his a d d e d m a s s a n d d a m p i n g coefficients. T h e solid line was computed using t h e double body mj's a n d the long dash curve results from using the simplified version of the mj's (32). A s c a n be s e e n , f o r this smooth s l e n d e r b o d y t h e different mj's do not change the results very m u c h .

The m o s t prominent feature of figure 6 is t h e singularity at x = 1/4. The strip theory results are not affected by x = 1/4 a n d the experiments w e r e not c o n d u c t e d at such low frequencies. A s previously discussed, t h e x = 1/4 singularity will affect all linear t h r e e - d i m e n s i o n a l theories. It will b e s h o w n in figures 9 - 1 3 that it also affects the body-nonlinear problem a n d random s e a simulations.

In figure 6 the agreement between t h e o r y a n d experiment is m i x e d . For pure h e a v e , t h e s t r i p theory a p p e a r s better while for the cross-couplings a n d for p u r e pitch t h e t h r e e d i m e n s i o n a l t i m e -domain predictions give better agreement. For this s h i p , u s i n g t h e double body mj's is not w o r t h t h e e x t r a e f f o r t . P e r h a p s the c o m p l e t e mj's m i g h t improve the a g r e e m e n t for all modes of motion. It s h o u l d be n o t e d t h a t in t h e b o d y - n o n l i n e a r c o m p u t a t i o n s t h e mj t e r m s a r e a u t o m a t i c a l l y accounted for. T h e agreement between strip theory, linear t i m e - d o m a i n a n a l y s i s , a n d e x p e r i m e n t s shown in figures 5 and 6 is typical cf the results that have b e e n calculated for many different ship t y p e s (see Magee a n d Beck, 1988).

As d i s c u s s e d previously, the c h a r a c t e r of t h e curves s h o w n in figures 5 and 6 around the irregular frequencies a n d at x = 1/4 is the result of the large-time tail in t h e t i m e domain. This is d e m o n s t r a t e d by figure 7 w h i c h s h o w s an e x p a n d e d view of the large-time tail of the pitch moment t i m e - d o m a i n record f o r t h e Wigley hull calculations. At a Froude number of 0.3 the tail is dominated by the x = 1/4 frequency a n d at zero fonA/ard s p e e d the irregular frequencies appear. Note that t l i e zero s p e e d tail has been multiplied by lOOx in order to plot it on t h e s a m e c u r v e as t h e f o r w a r d s p e e d result. T h e apparent beat f r e q u e n c y in the zero s p e e d tail i s ' probably the result of interference between the t w o lowest i r r e g u l a r f r e q u e n c i e s . W e h a v e s e e n no evidence of irregular frequencies in calculations at forward s p e e d . This might be numerical because of the dominance of the x = 1/4 frequency o r it might be the result of f o r w a r d speed effects; at t h i s point w e do not know.

E x a m p l e s of b o d y - n o n l i n e a r c a l c u l a t i o n s a r e s h o w n in figures 8-13. T h e results a r e all f o r a s u b m e r g e d ellipsoid of length-to-diameter ratio of 5. At the p r e s e n t t i m e , the computer c o d e f o r b o d y

0 . 3

-«

-•

*

1

>

i

Figure 6. Added mass and damping for a Wigley hull, Fn=0.3. — double-body mj's; Time domain, approximate mj's; + Experiments, Gerritsma, 1988.

Time domain, — Strip theory;

Figure 7. Large-time tail of the time-domain

record of the pitch moment for the Wigiey huil.

Fn=0.3; Fn=0.0.

FnsO.O solution has been expanded 100X.

0 . 0 2 0 " > Ü O L

/

1 1

1 1

Figure 8. Surge and heave force for a

submerged ellipsoid started from rest at two

different depths of submergence.

Time domain, H/L=0.16;

Doctors and Beck, 1987 H/L=0.16;

Time domain, H/L=0.245;

^ Doctors and Beck, H/L=0.245.

48.0

>

Figure 9. Greatly expanded view of the

approach to steady state for H/L=0.245.

Time domain;

Doctors and Beck.

nonlinear calculations in the case of floating bodies is not complete. In order to save computer time, the ellipsoid has b e e n a p p r o x i m a t e d by 36 panels on one-half of the body (12 lengthwise x 3 girthwise). T h i s is the s a m e configuration used by Doctors a n d B e c k (1987) for s o m e of t h e i r steady N e u m a n n -Kelvin calculations. Using more panels will alter t h e absolute value of the forces, but the character of the c u r v e s r e m a i n s t h e s a m e . For n u m e r i c a l investigations of a s u b m e r g e d body, the 36 panel ellipsoid seems sufficient. T h e calculations were all made starting from rest, using a smooth start-up in both forward s p e e d a n d h e a v e motion. T h e time histories of the f o r w a r d s p e e d a n d forced heave motion are given b y :

y ( r ) = [ / o f i - e - ^ ' ^ l (36)

Note that this m o t i o n has z e r o initial velocity a n d acceleration, but contains a jerk at t = 0.0.

F i g u r e 8 s h o w s t h e s u r g e f o r c e ( w a v e resistance) and heave force acting on the ellipsoid as it starts from rest a n d a p p r o a c h e s a c o n s t a n t f o r w a r d s p e e d . The ultimate Froude number is 0.35. T h e t w o depth to length ratios are 0.16 a n d 0.245. T h e straight lines are f r o m t h e steady N e u m a n n -Kelvin code used by Doctors a n d Beck (1987). The s a m e b o d y geometry w a s u s e d for both c o d e s , but t h e t i m e - d o m a i n results w e r e e x t r a p o l a t e d using three values of the time step size. As can be s e e n , t h e t i m e - d o m a i n results quickly approach the steady state v a l u e s . The large-time values for both heave a n d s u r g e a g r e e w i t h t h e s t e a d y N e u m a n n - K e l v i n results to within 0 . 5 % e v e n for the smaller depth of s u b m e r g e n c e . This is a g o o d verification for t w o entirely different c o m p u t e r c o d e s .

If figure 8 is e x a m i n e d very closely, it c a n be s e e n t h a t there are oscillations in the t i m e - d o m a i n r e s u l t s a s they a p p r o a c h s t e a d y state. Figure 9 s h o w s a greatly exaggerated scale of the approach to s t e a d y state for the surge a n d heave forces at the d e e p e r d e p t h . T h e s u r g e force is e x p a n d e d 125x a n d t h e heave force 400x. T h e oscillations are at a f r e q u e n c y e q u i v a l e n t t o T = 1/4 a n d d e c a y approximately a s 1/t. T h e s e results are in complete a g r e e m e n t w i t h W e h a u s e n ( 1 9 6 4 ) in w h i c h he investigated the effects of t h e starting transient of a thin s h i p started from rest. R g u r e 9 shows that even a s m o o t h start-up leads t o s o m e oscillations. T h e m o r e abrupt the start-up, the bigger the oscillations.

T o i n v e s t i g a t e t h e e f f e c t s of x = 1/4 o n s i m u l a t i o n s of sinusoidal m o t i o n s , figure 10 s h o w s t h e h e a v e force on t h e e l l i p s o i d as a function of t i m e . T h e b o d y has f o r w a r d s p e e d ( u l t i m a t e

F n = 0 . 3 5 ) a n d is also heaving sinusoidally at a single f r e q u e n c y c o r r e s p o n d i n g to x = 1/4. T h e amplitude of motion is A/ L = 0.085 a n d the m e a n depth of s u b m e r g e n c e is HQ/L = 0.245. In figure 10a, the total heave force with and without artificial damping a n d t h e contribution to the force from just the (1/r -l/r") ternis is shown. The curves for the total force with a n d without artificial damping are almost coincident. T h e (1/r - 1/r*) terms do not vary from cycle to cycle a n d are responsible for a large part of the force, particulariy on the bottom of the stroke. A s e x p e c t e d , t h e w a v e t e r m s m a k e t h e b i g g e s t contribution near the free surface.

The difference between the total force and the (1/r -l/r") component is shown in figure 10b. As can be s e e n , t h e p e a k s in the force curves are growing slowly for no artificial damping and reach constant amplitude for t h e c a s e with d a m p i n g . A line is drawn through the peaks of the curves to emphasize the growth without artificial damping. Since there is a singularity in t h e frequency domain response at X = 1/4, it is expected that the results without artificial damping will continue to grow and there will be no s t e a d y s t a t e s o l u t i o n . T h e artificial d a m p i n g apparently eliminates this growth. The growth rate for the case without artificial damping is extremely slow. Dagan a n d Ivliloh (1980) show that in three-dimensions the singularity in the frequency domain behaves in the vicinity of x = 1/4 as / /I | Ö J - Ü)C| , w h e r e cOc is the X = 1/4 critical frequency. While we have not yet w o r k e d out t h e asymptotic f o r m for t h e growth, it is probably on the order of in{t).

Figure 11 presents the surge force for the x = .20 case with the s a m e parameters as used for figure 10, e x c e p t that t h e frequency was lowered. No artificial d a m p i n g w a s u s e d in the c o m p u t a t i o n . Several interesting features of this figure should be noted. First, the peaks in the unsteady force do not g r o w as o p p o s e d t o the x = 1/4 case. T h e r e is an initial starting transient and then the peaks exhibit a small oscillatory behavior as they approach steady state. F o r otfier c o m b i n a t i o n s of frequency a n d forward s p e e d t h e oscillatory behavior can be much more p r o n o u n c e d . It is easily calculated that this oscillation is t h e beat frequency b e t w e e n x = 1/4 and X = . 2 0 . T h e starting transient must induce s o m e X » 1/4 response a n d this forms a beat with the X = .20 force. Also s h o w n in the figure is the a p p r o a c h to s t e a d y f o r w a r d s p e e d if t h e r e is no forced h e a v e (a replot of the curve in figure 8 as long d a s h e s at s m a l l times) and the m e a n of the total surge force n e a r the end of the record (small d a s h e s at large t i m e s ) . T h e vertical d i f f e r e n c e between t h e s e t w o lines is the mean a d d e d drag d u e to h e a v e . T h i s is another a d v a n t a g e of t h e body nonlinear time-domain method: the mean shift a n d s l o w l y v a r y i n g f o r c e s are a u t o m a t i c a l l y c o m p u t e d , no special calculations are necessary.

Ü Q . T V ( G / L )

>

Figure 10. (a) Heave force on a heaving ellipsoid at Fn=0.35, t=1/4, amplitude A/L=0.085, mean depth Ho/L=0.245.

0=0.0; — 0=0.05; (1/r - 1/r') force.

(b) Expanded view of total force minus (1/r - 1/r') force.

> O

Q .

T V ( G / L )

Figure 12. (a) Surge force on a heaving ellipsoid at Fn=0.35, with a sum of five sine waves In heave, (b) Surge force due to 4 sine

waves In heave. o=0.0;

0=0.05; DIfferenc between these two.

> O Q . - o . o i »•

A A A

Figure 11. Surge force on a heaving ellipsoid at Fn=0.35, x=0.20, A/L=0.085, H o / L = 0 . 2 4 5 ;

Total;

Forward speed only; Mean.

Figure 13. Greatly expanded view of the difference forces from Figure 12.

With T=1/4; Without t=1/4.

T h e e f f e c t s of x = 1/4 o n m o r e g e n e r a l simulations are s h o w n in figures 12 and 13. T h e ' a n d o m s e a simulations were m a d e using a s u m of l i v e sine w a v e s in h e a v e a s in (37) w i t h f o r w a r d s p e e d . Figure 12a has o n e c o m p o n e n t p l a c e d at C a 1/4 a n d in 12b t h i s c o m p o n e n t h a s b e e n sliminated. E a c h of the simulations were run with a n d without t h e artificial d a m p i n g . T h e d a s h e d line hear the axis is the difference between the two runs. Without t h e x = 1/4 c o m p o n e n t (figure 12b) t h e difference is so small it cannot be seen in the figure. For the x = 1/4 c a s e , t h e difference c o n t i n u e s to g r o w . Initially, t h e d i f f e r e n c e is e x a c t l y z e r o because t h e artificial d a m p i n g affects only the large t i m e a s y m p o t i c f o r m of t h e t i m e - d o m a i n G r e e n function. A greatly expanded view of the differences s s h o w n in figure 13. A s can be s e e n , there are some d i f f e r e n c e s b e t w e e n the artificially d a m p e d a n d u n d a m p e d c a l c u l a t i o n s e v e n f o r t h e c a s e A/ithout t h e x = 1/4 c o m p o n e n t ; h o w e v e r , t h e y remain very s m a l l . W i t h the x = 1/4 c o m p o n e n t Dresent, t h e differences are m u c h larger a n d m o r e mportantly, they continue to g r o w . It appears t h a t a n y long-time simulation which uses the linear free surface b o u n d a r y c o n d i t i o n a n d i n c l u d e s x = 1/4 forcing is d o o m e d to failure because this c o m p o n e n t will eventually dominate the solution. Our p r o p o s e d artificial d a m p i n g f i x - u p is s i m p l e but e f f e c t i v e . Undoubtedly, improved methods can be d e v e l o p e d . 5. C O N C L U S I O N S

T h e primary conclusion from t h e work to d a t e on t i m e - d o m a i n analysis is that it is a viable alternative to t h e traditional f r e q u e n c y - d o m a i n a n a l y s i s . For

i n e a r i z e d p r o b l e m s w i t h f o r w a r d s p e e d , : o m p u t a t i o n s in the time d o m a i n appear to be m u c h a a s i e r t h a n t h e e q u i v a l e n t f r e q u e n c y - d o m a i n Dalculations. T h e W i g l e y m o d e l six d e g r e e s of f r e e d o m c a l c u l a t i o n s f o r l i n e a r r a d i a t i o n a n d sxciting force c o e f f i c i e n t s u s i n g 240 p a n e l s o n a l a l f - b o d y a n d 2 5 6 t i m e s t e p s r e q u i r e d 2 5 C P U •ninutes for Fp = 0.3 on a C R A Y Y-fvlP. For radiation 'orces only, at zero forward s p e e d the required t i m e ivas 9 C P U m i n u t e s . I n c o m p a r i s o n w i t h a x p e r i m e n t s a n d s t r i p t h e o r y , l i n e a r i z e d t i m e -domain analysis g i v e s m i x e d results. At t i m e s t h e t i m e - d o m a i n c a l c u l a t i o n s c o m p a r e b e t t e r w i t h axperiments and at other t i m e s t h e predictions are worse. For the mathematical W i g l e y hull f o r m , t h e j s e of d o u b l e body or simplified m^s does not s e e m to m a k e m u c h of a difference in t h e a d d e d m a s s a n d damping predictions.

T i m e - d o m a i n analysis is easily extended to t h e D O d y - n o n l i n e a r p r o b l e m i n w h i c h t h e b o d y soundary condition is always satisfied on t h e e x a c t n s t a n t a n e o u s w e t t e d s u r f a c e of body. R e s e a r c h nto t h e body-nonlinear problem is continuing at T h e J n i v e r s i t y of Michigan. T o date computations h a v e seen p e r f o r m e d only o n s u b m e r g e d bodies.

Calculations w i t h a s u b m e r g e d ellipsoid have s h o w n e x c e l l e n t a g r e e m e n t w i t h a s t e a d y Neumann-Kelvin c o d e . In addition, the calculations h a v e indicated t h e i m p o r t a n c e of t h e f r e q u e n c y d o m a i n singularity at x = 1/4. Because of the linear f r e e s u r f a c e b o u n d a r y c o n d i t i o n , s i n g u l a r i t i e s a p p e a r in the frequency d o m a i n results at x = 1/4 a n d the results g r o w slowly without bound in time-d o m a i n simulations. A simple fix-up using artificial d a m p i n g on the large-time tail of the time-domain G r e e n f u n c t i o n is p r o p o s e d . In t i m e - d o m a i n s i m u l a t i o n s , the fix-up p r e v e n t e d t h e u n b o u n d e d g r o w t h a n d othenwise d i d not alter the solution significantly. Much more research needs to be done on t h e large-time asymptotics and rates of decay in order to obtain the exact solutions in both the time a n d frequency domains.

A C K N O W L E D G E M E N T S

This research was funded by The Office cf Naval R e s e a r c h , C o n t r a c t N o . N 0 0 0 1 4 - 8 8 - K - 0 6 2 8 . C o m p u t a t i o n s w e r e m a d e in part using a C R A Y G r a n t , U n i v e r s i t y R e s e a r c h a n d D e v e l o p m e n t Program at the San Diego Supercomputer Center. R E F E R E N C E S

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